Abstract
We investigate the local interior regularity condition of a suitable weak solution to 3D MHD equations. We prove that if the gradient of a velocity vector belong to a local BMO space \(\mathrm{bmo}_r\) in a neighborhood of an interior point, then solution is regular near that point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chang, D.-C.: The dual of Hardy spaces on a bounded domain in \(\mathbb{R} ^n\). Forum Math. 6, 65–81 (1994)
Chae, D., Kang, K., Lee, J.: On the interior regularity of suitable weak solutions to the Navier–Stokes equations. Commun. P.D.E. 32, 1189–1207 (2007)
Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)
Galdi, G.P.: In: Galdi, G.P., Heywood, J., Rademacher, R. (eds.) An Introduction to the Mathematical Theory of the Navier. Stokes Initial-Boundary Value Problem, vol. 1. Birkhauser-Verlag, Basel (2000)
Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)
Grujić, Z.: Localization and geometric depletion of vortex-stretching in the 3D NSE. Commun. Math. Phys. 290, 861–870 (2009)
Grujić, Z., Guberović, R.: A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations. Ann. Inst. H. Poincare Anal. Non Lineaire. 27, 773–778 (2010)
He, C., Xin, Z.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J. Funct. Anal. 227, 113–152 (2005)
Ladyžhenskaya, O.A., Solonnikov, V.A.: Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid. Proc. V.A. Steklov Math. Inst. 59, 115–173 (1960). (in Russian)
Neustupa, J., Pokorny, M.: An interior regularity criterion for an axially symmetric suitable weak solution to the Navier–Stokes equations. J. Math. Fluid Mech. 2, 381–399 (2000)
Sadek, G.: Regularity criterion on weak solutions to the Navier–Stokes equations. J. Korean Math. Soc. 45, 537–558 (2008)
Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)
Vyalov, V.: Partial regularity of solutions to the magnetohydrodynamic equations. J. Math. Sci. (N. Y.) 150, 1771–1786 (2008)
Yun, W.: BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations. J. Math. Anal. Appl. 328, 1082–1086 (2007)
Acknowledgements
Funding was provided by National Research Foundation of Korea (Grant Nos. NRF-2020R1C1C1A01006521).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author states that there is no conflict of interest.
Additional information
Communicated by S. Friedlander.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, JM. A Regularity Criterion for the 3D MHD Equations in a Local BMO Space. J. Math. Fluid Mech. 24, 53 (2022). https://doi.org/10.1007/s00021-022-00683-6
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-022-00683-6